The shape of the lesson
By the end of the lesson, learners can
- Lines become boundaries: In the lab, type an inequality such as \(y < 2x + 1\).
- Which side?: For a sloping boundary, pick a test point that is not on the line, often \((0, 0)\).
- Many inequalities, one region: Draw every boundary, shade the unwanted side of each, and the region R that stays clear satisfies all the inequalities at once.
- Line: change the sign to \(=\) and draw the line (use intercepts or \(y = mx + c\)).
- Type: solid for \(\le, \ge\); broken for \(<, >\).
- Side: test a point not on the line, usually \((0, 0)\).
- Shade: the unwanted side, leaving the region R clear.
- Integer points: go row by row, and check points on each boundary against its sign.
One picture every idea returns to
Draw the boundary as if it were an equation: SOLID for \(\le\) or \(\ge\), BROKEN for \(<\) or \(>\). Test a point that is not on the line to decide which side is wanted, then shade the side you do NOT want.
Forty-five minutes, phase by phase
| Time | Phase | What happens in the room | Grouping |
|---|---|---|---|
| 0 to 5 min | Hook: predict | Open with the question in the title, Shade what you don't want.. Extended. Put the figure on the board: it is the one picture every idea in the lesson returns to. | Whole class, then pairs |
| 5 to 13 min | Teach and check: Lines become boundaries | In the lab, type an inequality such as \(y < 2x + 1\). Project tab 6 of the lab (Equations and inequalities lab, regions) and let a student drive it while the class predicts. Model the worked example Vertical and horizontal lines on the board, one line at a time. | Whole class, then pairs |
| 13 to 21 min | Teach and check: Which side? | For a sloping boundary, pick a test point that is not on the line, often \((0, 0)\). Model the worked example Test the origin on the board, one line at a time. Your turn: Does the region \(x + 2y > 8\) contain the origin? | Pairs |
| 21 to 30 min | Teach and check: Many inequalities, one region | Draw every boundary, shade the unwanted side of each, and the region R that stays clear satisfies all the inequalities at once. Model the worked example Integer points in a region on the board, one line at a time. Your turn: List the integer points that satisfy \(x \ge 1\), \(y \ge x\) and \(x + y < 6\). | Connect-Extend-Challenge |
| 30 to 36 min | Check | The four Stage 1 questions from the student lesson, answered on mini whiteboards or by show of fingers. Re-teach the one most get wrong. | Individual, then whole class |
| 36 to 43 min | Plenary | The exam-style questions from Stage 2, one at a time: thirty seconds alone, then a random call. The feedback lines in the key give the examiner's wording. | Individual, random call |
| 43 to 45 min | Exit and homework | Set the worksheet below (6 questions, 17 marks). Learners self-assess against the objectives before they leave. | Individual |
Protect the parts that carry the learning
The sequence carries about fifty minutes of material, so the lab is driven by one student while the class predicts rather than explored freely. Times are a guide.
Protected: the check at 30 minutes. It is the only individual read on whether the key idea landed, and it is not cut.
Cut first if the build overruns: the last check in Many inequalities, one region, never the check or the plenary.
In a 60 minute block: run worksheet questions 1 to 3 in class as a Rally Coach set and model both the hardest worked example and its check live.
Connect-Extend-Challenge
After the new idea, each student writes three lines: how it connects to what they knew, how it extends their thinking, and what still challenges them. Pairs compare, then the challenges become the plenary.
The full structure, with its variations and the other twelve, is on the cooperative-learning page.
What to head off, and how
| Trap learners fall into | Teaching move that pre-empts it |
|---|---|
| \(x = 3\) is a vertical line, not a horizontal one. | Students often draw \(x = 3\) across the page because '3 on the \(x\)-axis' feels horizontal. A line \(x = k\) is always vertical; \(y = k\) is always horizontal. |
| If the line passes through \((0, 0)\), such as \(y < 2x\), the origin cannot be a test point: it is ON the line. | Use \((1, 0)\) instead: \(0 < 2\) is true, so the side containing \((1, 0)\) is wanted. |
| Points on a broken line are NOT in the region; points on a solid line ARE. | Check every boundary point against the sign before listing it. |
Inequalities as regions on a graph: worksheet
6 questions, 17 marks. Written in the style of the papers; no past paper question is reproduced.
Write down the inequality for (a) the region to the left of the solid line \(x = 3\), (b) the region above the broken line \(y = -2\).
State whether each boundary line is solid or broken. (a) \(y \ge 2x\) (b) \(x + y < 5\)
Do the points (a) \((3, -1)\) and (b) \((4, 0)\) satisfy \(2x - y > 7\)?
A broken line passes through \((0, 2)\) and \((4, 0)\). The wanted region is the side NOT containing the origin. Find the inequality.
The region R satisfies \(x \ge 1\), \(y \ge x\) and \(x + y < 6\). List all the points in R with integer coordinates.
A club hires \(x\) small buses and \(y\) large buses. A small bus seats 20 people and a large bus seats 40. At least 120 seats are needed, no more than 5 buses can be hired, and at least one of each type is hired.
(a) Show that \(x + 2y \ge 6\). (b) Write down the other inequalities. (c) List every possible combination \((x, y)\).
Answer key and marking notes
- (a) \(x \le 3\)
- (b) \(y > -2\)
- (a) Solid, because \(\ge\) includes the line.
- (b) Broken, because \(<\) excludes the line.
- (a) \(2(3) - (-1) = 7\), and \(7 > 7\) is false. No: the point is on the boundary, which is excluded.
- (b) \(2(4) - 0 = 8 > 7\). Yes.
- Gradient \(= \dfrac{0 - 2}{4 - 0} = -\dfrac{1}{2}\), so \(y = 2 - \dfrac{1}{2}x\), i.e. \(x + 2y = 4\).
- The origin gives \(0 < 4\), so the origin's side is \(x + 2y < 4\).
- The wanted side is the other one, and the line is broken: \(x + 2y > 4\).
- \(x = 1\): \(y \ge 1\) and \(y < 5\), so \(y = 1, 2, 3, 4\).
- \(x = 2\): \(y \ge 2\) and \(y < 4\), so \(y = 2, 3\).
- \(x = 3\): \(y \ge 3\) and \(y < 3\), none.
- \((1,1), (1,2), (1,3), (1,4), (2,2), (2,3)\)
- (a) Seats: \(20x + 40y \ge 120\). Divide by 20: \(x + 2y \ge 6\).
- (b) \(x + y \le 5\), \(x \ge 1\), \(y \ge 1\).
- (c) \(y = 1\): \(x \ge 4\) and \(x \le 4\), so \((4, 1)\).
- \(y = 2\): \(x \ge 2\) and \(x \le 3\), so \((2, 2), (3, 2)\).
- \(y = 3\): \(x \ge 1\) and \(x \le 2\), so \((1, 3), (2, 3)\). \(y = 4\): \(x = 1\), so \((1, 4)\).
- Six combinations: \((4,1), (2,2), (3,2), (1,3), (2,3), (1,4)\).
6 multiple-choice questions on this lesson
From the topic question bank. Assign them in the app for a marked score in your gradebook, or run the topic as a class game.
Which boundary line should be drawn to show the region \(y \ge 2x - 3\)?
- AA solid line \(y = 2x - 3\)
- BA broken line \(y = 2x - 3\)
- CA solid line through \((0, 3)\) with gradient 2
- DA broken line through \((0, -3)\) with gradient \(-2\)
Which point lies in the region defined by \(x + 2y < 10\) and \(y > x\)?
- A\((1, 4)\)
- B\((2, 4)\)
- C\((3, 2)\)
- D\((4, 5)\)
The region R is bounded by the solid line \(y = 1\), the broken line \(x = 4\) and the solid line \(y = x + 1\). Which three inequalities define R?
- A\(y \ge 1,\ x < 4,\ y \le x + 1\)
- B\(y \ge 1,\ x \le 4,\ y \le x + 1\)
- C\(y \ge 1,\ x < 4,\ y \ge x + 1\)
- D\(y > 1,\ x < 4,\ y \le x + 1\)
How many points with integer coordinates satisfy all of \(x > 0\), \(y \ge 1\) and \(2x + y \le 8\)?
- A12
- B9
- C16
- D20
A solid line passes through \((0, -2)\) and \((4, 0)\). The wanted region is the side of the line containing the origin. Which inequality describes it?
- A\(x - 2y \le 4\)
- B\(x - 2y \ge 4\)
- C\(x - 2y < 4\)
- D\(2y - x \le 4\)
Which point lies in the region \(y > 3x\)?
- A\((1, 4)\)
- B\((1, 3)\)
- C\((2, 5)\)
- D\((0, 0)\)
Quiz key
The boundary is \(y = 2x - 3\) (gradient 2, \(y\)-intercept \(-3\)). The sign \(\ge\) includes points on the line, so the line is SOLID. A broken line is only for strict signs \(<\) and \(>\); the option through \((0, 3)\) has the wrong intercept sign.
\((1, 4)\): \(1 + 8 = 9 < 10\) and \(4 > 1\), both true. \((2, 4)\) gives \(2 + 8 = 10\), which is ON the broken line \(x + 2y = 10\), so it is excluded. \((3, 2)\) fails \(y > x\); \((4, 5)\) gives 14, not less than 10.
Test \((3, 2)\): \(2 \ge 1\), \(3 < 4\) and \(2 \le 3 + 1\). Solid lines take \(\le\) or \(\ge\), the broken line takes a strict sign. So \(y \ge 1,\ x < 4,\ y \le x + 1\). \(x \le 4\) ignores that \(x = 4\) is broken; \(y \ge x + 1\) is the region above the sloping line, not below it.
\(x = 1\): \(y \le 6\), so \(y = 1\) to 6 (6 points). \(x = 2\): \(y \le 4\) (4 points). \(x = 3\): \(y \le 2\) (2 points). \(x = 4\): \(y \le 0\), none. Total 12. Making \(2x + y \le 8\) strict gives 9, which wrongly drops points on the SOLID line; using \(y \ge 0\) gives 16; using \(x \ge 0\) gives 20.
Gradient \(= \dfrac{0 - (-2)}{4 - 0} = \dfrac{1}{2}\), so \(y = \dfrac{1}{2}x - 2\), which is \(x - 2y = 4\). The origin gives \(0 - 0 = 0 \le 4\), true, and the line is solid, so \(x - 2y \le 4\). \(x - 2y < 4\) has the right side but the wrong line style; \(2y - x \le 4\) describes a different line, \(y = \dfrac{1}{2}x + 2\).
\((1, 4)\): \(4 > 3\), true. \((1, 3)\) and \((0, 0)\) make \(y = 3x\) exactly, so they lie ON the broken boundary and are excluded. This is also why the origin cannot be used as a test point for this line. \((2, 5)\) gives \(5 > 6\), false.
What to project
The student lesson embeds these tabs. Open them full screen on the projector; every slider has a drag handle on the stage.
Slides for this lesson
A projectable deck for every pack is on the way. Until then, present the student lesson full screen: its sections, figure and lab tabs follow the same order as the plan above.